PEAK FACTOR FOR NON-GAUSSIAN PROCESSES REVISITED

Transcription

1 PEAK FACTOR FOR NON-GAUSSIAN PROCESSES REVISITED Dae-Kun Kwon 1 and Ahsan Kareem 2 1 Postdoctoral research associate, NatHaz Modeling Laboratory, University of Notre Dame, dkwon@nd.edu 2 Robert M. Moran Professor, NatHaz Modeling Laboratory, University of Notre Dame, kareem@nd.edu ABSTRACT This paper discusses non-gaussian peak factor for univariate stationary non-gaussian processes, which can be used for estimating expected positive and negative extremes of non-gaussian processes, e.g., a time history of wind pressure fluctuations in separated flow regions like the roof of a low-rise building. A previous analytical expression for the peak factor for non-gaussian processes given in [Kareem and Zhao (1994)] based on a moment-based Hermite model is revisited and the standard deviation of this estimate is derived. In addition, some improvements concerning the estimation of the peak factor are discussed. Examples illustrate the efficacy of this data-driven model-based peak factor approach. KEYWORDS: PEAK FACTOR, NON-GAUSSIAN PROCESS Introduction The assurance of safety and reliability of structures requires estimation of the extremes of the applied load effects. If the statistical description of input and/or output in a system differs significantly from Gaussian, conventional methodologies with implicit assumption of Gaussianity may no longer be valid requiring a non-gaussian estimation framework. This is particularly important when considering the extremes, which are sensitive to the tail regions of the probabilistic description of non-gaussian processes. Conventional peak factors for Gaussian processes [Davenport (1964)] have been widely used in most codes and standards to estimate the expected extremes of wind loads and the related response in design procedures. However, this factor generally yields nonconservative values when applied to non-gaussian processes. To overcome this shortcoming, [Kareem and Zhao (1994)] proposed analytical form of a univariate non- Gaussian peak factor. Despite its usefulness in predicting extremes of non-gaussian processes, there remains some ambiguity regarding robustness of its application due to a lack of consistent description of some of the parameters involved. This has led to a few studies that have in correctly applied this with misleading conclusions. In this study, a non-gaussian peak factor by [Kareem and Zhao (1994)] is revisited to suggest its corrected expression and demonstrated its effectiveness with subtle improvements. In addition, standard deviation of the non-gaussian peak factor estimate is derived, which, due to a large variability in the peak value estimates, is believed to be more significant than that of its counterpart Gaussian processes. Non-Gaussian Peak Factor [Kareem and Zhao (1994)] proposed an analytical form of the univariate non-gaussian peak factor (g ng ), which was based on the concept of a translation process [Grigoriu (1984)], moment-based Hermite model [Winterstein (1988)] and the framework of Gaussian peak factor [Davenport (1964)]. Several places in the literature it has been misunderstood as a gust factor. Note that the original formula in [Kareem and Zhao (1994)] has been subsequently utilized in later publications by the authors [Gurley et al. (1997)], [Kareem et al. (1998)], which lacked consistency as some parameters were defined alternatively and a few minor variations in both the expressions and variables have been noted. Thus, for the sake of clarity, a consistent version of the non-gaussian peak factor with proper description of variables

2 involved is presented here to preclude any further misinterpretation of the expression proposed originally in [Kareem and Zhao (1994)]: 2 γ π g = κ β + + h β + 2γ 1+ + h β + 3β 3 4 ( γ 1 ng ) + γ + γ β β (1) β 6 β where, x = a non-gaussian process; y = a standardized non-gaussian process obtained from a process x, i.e., y = (x E[x])/σ x ; h 3, h 4, κ = coefficients of moment-based Hermite model [Winterstein (1988)]; γ = Euler s constant ( ); ν 0 = mean zero upcrossing rate of a process y; m i = i th spectral moment of a process y; S y (n) = one-sided power spectral density of a process y; n = frequency in Hertz; T = duration; γ 3 = skewness of a process y; γ 4 = kurtosis (here, normalized 4 th central moment 3, so-called, excessive kurtosis) of a process y; m 2 i β = 2ln ( ν0t), ν0 =, mi = n S ( ) 0 y n dn m 0 (2) γ γ h3 =, h4 =, κ = γ 1+ 2h + 6h For practical applications, it is recommended that the original process x be standardized as process y with a zero mean and a unit variance. Accordingly, the coefficients in Eq. (2) are computed based on the standardized process y only. In addition, if a process x is Gaussian which results in y being Gaussian, then the non-gaussian peak factor reduces to conventional one [Davenport (1964)], since both γ 3 & γ 4 become zero that leads h 3 & h 4 to become zero and κ = 1 (Eq. 7b). In this manner, conventional Gaussian peak factor proposed by [Davenport (1964)] may also be viewed as a special case of non-gaussian peak factor given in Eq. (1). Finally, reminding that the g ng is indeed the expected value of the extreme of a process y, therefore, the extreme of a process x should be calculated from the following expression. xext = x + gng σ x ( positive extreme) (3) xext = x gng σ x ( negative extreme) It is also worth noting that Eq. (1) was formulated for a positive extreme, thus for a negative extreme, opposite sign of γ 3, i.e., -γ 3 needs to be introduced to estimate the negative non- Gaussian peak factor, e.g., a time history of negative wind pressure coefficients at the roof corner of a low-rise building or corners along faces of a tall building. It is envisaged that this could help to alleviate any ambiguity surrounding the peak factor. Standard Deviation of Non-Gaussian Peak Factor Unlike Gaussian processes where it is sufficient to assume that the extremes are equal to the mean extreme as the variability of the extreme distribution is believed to be small [Davenport (1964)], for non-gaussian process, the variability of the peak is usually large in either a positive or negative tail and the observed peak may differ significantly from its mean value also noted in [Sadek and Simiu (2002)]. This led to the motivation of deriving an expression for the standard deviation of non-gaussian peak factor (σ ng ). Following the framework for Gaussian processes by [Davenport (1964)], σ ng has been derived in this study to be equal to: 2 π h3 σng κ h h4 1.64β β β β (4) 1/ h h 2 + 3h4 1.64β + β β

3 The level of σ ng may provide guidance towards establishing appropriate measures of non- Gaussian processes. Akin to the characteristics of non-gaussian peak factor formula in Eq. (1), this standard deviation (Eq. 4) reduces to π / 6β for Gaussian processes as h 3, h 4 = 0 and κ = 1, which is identical to conventional one [Davenport (1964)]. In addition, the standard deviation of the mean extreme in a process x is calculated from the following expression: σ = σ σ (5) x, ext ng x Improvements of Non-Gaussian Peak Factor Estimation Although the non-gaussian peak factor formulation given in Eq. (1) is based on consistent theoretical background, one problem remains that it may yield slightly conservative value when the process exhibits strong non-gaussianity, e.g., relatively large skewness and/or kurtosis. This is due to the nature of the moment-based Hermite model [Winterstein (1988)] that was derived based on the assumption of small deviations from Gaussianity. In order to overcome this shortcoming, two possible improvements are discussed here. Modified Hermite Model In [Tognarelli et al. (1997)] and [Gurley et al. (1997)], a modified Hermite model was presented that facilitated an improvement in the evaluation of a non-gaussian process y by the Hermite model of polynomial coefficients (h 3 & h 4 ) in the following equations γ3= κ ( 8h h3h4 + 36h3h4 + 6h3) (6) γ4 + 3 = κ ( 60h h h3h4 + 60h h h h3h4 + 3) These equations were designed simply to obtain nonlinear solutions of h 3 & h 4 by setting γ 3 & γ 4 to be equal to those of the standardized process y, then these new h 3 & h 4 values obtained from Eq. (6) are used to compute g ng [Eq. (1)] in lieu of their original expressions in Eq. (2). This has been demonstrated to improve the accuracy of the Hermite model and PDF estimation of the standardized process y. Additional details are available in [Gurley and Kareem (2001)]. Revised Hermite Model [Winterstein et al. (1994) and Winterstein & Kashef (2000)] proposed simple expressions for h 3 & h 4 based on optimal results that minimize a lack-of-fit errors in skewness and kurtosis of the Hermite model in [Winterstein (1988)]. These values are intended to apply for the ranges of γ 3 & γ 4 : 0 < γ 4 < 12; 0 γ 2 3 < (2γ 4 )/3, which may include most cases of practical interest: γ 4 1/3 γ γ γ γ 3 [ γ 4 ] 1 h3 =, h4 = h40 1, h40 = (7) γ4 γ4 10 In comparison with the modified Hermite model, this form may be more convenient to use due to its analytical closed-form in lieu of solving coupled nonlinear equations in Eq. (6), if its performance is equivalent to the modified Hermite model. Examples Measured Pressure Fluctuations on a Full-Scale Low-Rise Building To investigate non-gaussian peak factor and its prediction based on the improvements, an example concerning a time history of the measured pressure fluctuations on a full-scale low-rise building [Thomas et al. (1995)] as shown in Fig. 1(a) are utilized. These fluctuations exhibit quite a large deviation from Gaussian (Table 1). In Fig. 1(b), a histogram of the pressure data, the probability density function (PDF) of the Gaussian fit and the PDF based on

4 Hermite model (HM) in Eq. (2) along with the Modified Hermite model (MHM) in Eq. (6), the Revised Hermite model (RHM) in Eq. (7), and a zoomed depiction of the negative tail region in the inset are shown. Overall, PDFs based on the MHM and RHM show better fits to histogram than the Gaussian fit and the HM, and their corresponding statistical characteristics compare better with given data (Table 1). If this data were assumed to follow Gaussian process, the Gaussian peak factor and its standard deviation are equal to and 0.332, which are quite smaller than the corresponding values of g ng and σ ng given by these models. This fact reinforces that conventional Gaussian peak factor cannot capture strong non- Gaussian features, therefore, Gaussianity should not be tacitly assumed in such cases. Comparing with the measured data, both MHM and RHM show a good comparison and which also have a smaller standard deviation in comparison with the HM. Fig. 1 (a) left: Measured pressure data; (b) right: Histogram of data and PDF estimations Table 1. statistical values, peak factors and its standard deviations by data and models σ y γ 3 γ 4 g ng σ ng data a HM MHM RHM a observed negative extreme of data Time Histories of TLP Sway and Wind Pressure on the Roof of Low-Rise Building Model Two types of measured time histories with standardization are illustrated here: first, sway response in tension leg platform (TLP) exhibiting mild non-gaussianity [Fig. 2(a); Table 2] and second, wind pressure coefficients in the roof of low-rise building model by wind tunnel test [Fig. 2(b); Table 2], which both data are used for estimating negative extremes in this study. In the case of TLP sway response, all three models (HM, MHM and RHM) exhibit almost the same results each other because given data slightly deviates from Gaussian, thus HM may be acceptable in this case, however, MHM & RHM show even better PDF estimates in view of statistics in data (Table 2) that are believed to be more accurate in estimates of both g ng and σ ng. On the contrary, the wind pressure coefficients by wind tunnel test [TPU: Aerodynamic database of low-rise building] in Fig. 2(b) exhibits strong non- Gaussianity, which results in discrepancies among HM and MHM/RHM. The observed trends are similar to the previous example.

5 Comparison with windpressure (NIST) The windpressure software [NIST (2005) and Main and Fritz (2006)] is designed to facilitate more widespread use of the database assisted design approach for low-rise buildings. The aerodynamic database is assembled by the U.S. National Institute of Standards and Technology (NIST), containing measured pressure time series for a fairly large number of gable-roofed building models with various dimensions contributed by the boundary layer wind tunnel laboratory, University of Western Ontario [Ho et al. (2003)]. This standalone MATLAB-based software has also employed the translation process model to estimate the extremes [Gioffrè et al. (2000) and Sadek and Simiu (2002)]. Internal force time series (e.g., bending moment) obtained from wind pressure coefficient data by wind tunnel test in conjunction with influence function of structure are fitted to Gamma distribution (for longer tail) and Gaussian distribution (for shorter tail). The extreme CDF is estimated by invoking mapping procedure, in which its PDF, differentiation of the CDF, is utilized to estimate the expected extreme value [Sadek and Simiu 2002]. Accordingly, comparison with the windpressure results offers another data set for verifications and insights on analytical non-gaussian peak factor approach employed in this study. Here, one sample of building and pressure database provided by NIST website [NIST 2005] is utilized with windpressure: W = 120 ft, L = ft, H = 18 ft, R = 5ft, Open Country. The windpressure conveniently provides with not only saving functionality for selected time series of calculated bending moments in test model, but graphically showing their estimated positive and negative extremes, which extreme values can be found via zooming of plots, owing to MATLAB s functionality [The Mathworks, Inc.]. Thus, two time histories of bending moments in frame 1 from windpressure, which can be treated as a univariate problem, are utilized for the comparison of estimated mean extremes in non- Gaussian peak factor and windpressure shown in Table 3: Case 1) at left knee, wind direction A = 15 (Fig. 3a); Case 2) at ridge, wind direction C = 165 (Fig. 3b). Overall, estimated extremes by non-gaussian peak factor in terms of MHM or RHM show great agreements with those from windpressure, while the factor with HM exhibits relatively conservative extremes, but good agreement shows in Case 2 where its non-gaussianity is regarded as mild, i.e., very small γ 3, γ 4 : this agrees with the underlying assumption of HM, as observed in the example of TLP sway response earlier (Table 2). Note that one discrepancy is observed in the negative extreme (or realistically a lower extreme) of Case 1, as the windpressure approach uses a Gaussian distribution to estimate its extreme. It is worth noting that although both this study and the windpressure employ the concept of the translation process [Grigoriu (1984)], analytical non-gaussian peak factor and its standard deviation expressions formulas offer more convenient format than windpressure procedure. windpressure employs the Gamma and Gaussian distributions to establish PDF of process y, which inevitably results in attendant parameter estimations such as probability plot correlation coefficient method [Sadek and Simiu (2002)]. While, this study employs moment-based Hermite model with improvement techniques (MHM or RHM) to analytically obtain PDF of a standardized process y, which the model shows very good PDF approximation up to the fourth moment as observed in earlier examples that simplifies the overall procedure. Furthermore, windpressure utilizes point-to-point mapping procedure to estimate an extreme CDF value, thus requiring iterations to generate the overall CDF are inevitable. In addition, extreme PDF is obtained from the numerical differentiation of the extreme CDF, consequently mean extreme and its standard deviation are computed in terms of the PDF by way of numerical integration. However, the model-based approach in this study can directly derive analytical forms of extreme CDF and PDF, which preferably lead to analytical forms of g ng and σ ng, thus offering more intuitive and convenient manner to estimate the expected non-gaussian extreme and its standard deviation in practical

6 applications, i.e., simply using analytical closed forms of g ng and σ ng without following procedures employed in windpressure. Fig. 2. Standardized time histories: (a) left: TLP sway response; (b) right: wind pressure coefficients from wind tunnel test (TPU) Table 2. Statistical values, peak factors and standard deviations from data and model-based approach [TLP sway response and wind pressure coefficients (TPU)] σ y γ 3 γ 4 g ng σ ng data a TLP Sway HM MHM RHM data a TPU data HM MHM RHM a observed negative extremes from standardized data Fig. 3. Time histories of bending moments in test model [NIST 2005]: (a) left: moment at left knee of frame 1, wind direction A = 15 ; (b) right: moment at ridge of frame 1, wind direction C = 165

7 Table 3. Comparison of estimated extremes by model-based approach and windpressure Standardized process Original process Pos. extreme c Neg. extreme c σ y γ 3 γ 4 peak std. peak std. data d 4.62 d Case 1 a HM MHM RHM data d d Case 2 b HM MHM Conclusions RHM a Frame 1, Response 1 : moment at left knee, wind direction A = 15 b Frame 1, Response 3 : moment at ridge, wind direction C = 165 c Estimated extremes and corresponding standard deviations of bending moments [units in lb-ft] d Results from windpressure [Main and Fritz (2006)] The non-gaussian peak factor and its standard deviation are discussed and their efficacy is demonstrated utilizing several data sets. The extremes of non-gaussian process associated with winds may not be accurately estimated by conventional peak factor, especially when a process exhibits strong non-gaussianity, which has been observed in a number of observations in the field of wind/structural engineering, e.g., wind pressures and corresponding internal forces in low-rise buildings, where roof damage is very prevalent. Furthermore, these models not only provide the expected value of the extreme, but also its associated standard deviation and PDF/CDF, which may have immediate applications in areas such as simulation, fatigue estimates and reliability analysis. It is worth noting that depending on the characteristics of the data being analyzed, extreme value estimates obtained through these models show a small deviation around the observed values due to the random nature of estimated quantity. The extreme value estimates derived simply from data analysis may not be as robust and reliable as those derived from data-driven models such as moment-based Hermite models. Acknowledgements The support for this study was, in part, provided by the NSF (Grant #CMS ) and the Global Center of Excellence at Tokyo Polytechnic University, Tokyo funded by MEXT. References Davenport, A. G. (1964). Note on the distribution of the largest value of a random function with application to gust loading. J. Inst. Civ. Eng., 24, Grigoriu, M. (1984). Crossings of non-gaussian translation processes. J. Eng. Mech., ASCE, 110(4), Gurley, K. R., Tognarelli, M. A., and Kareem, A. (1997). Analysis and simulation tools for wind engineering. Prob. Engng. Mech., 12(1), Gurley, K. R., and Kareem, A. (2001). Modeling of PDFs of non-gaussian system response. Proc. 8 th Int. Conf. on Str. Safety and reliability, Kyoto, Japan. Ho, T. C. E., Surry, D., and Nywening, M. (2003). NIST/TTU cooperative agreement - windstorm mitigation initiative: further experiments on generic low buildings, BLWT-SS , Phase 2 report. Kareem, A., and Zhao, J. (1994). Analysis of non-gaussian surge response of tension leg platforms under wind loads. J. offshore Mech. and Arctic Eng., ASME, 116,

8 Kareem, A., Tognarelli, M. A., and Gurley, K. R. (1998). Modeling and analysis of quadratic term in the wind effects on structures. J. Wind Eng. Ind. Aerodyn., 74-76, Main, J. A., and Fritz, W. P. (2006). Database-assisted design for wind: concepts, software, and examples for rigid and flexible buildings. NIST Building Science Series 180, NIST. National Institute of Standards and Technology (2005), windpressure - Database-Assisted-Design software for rigid, gable-roofed building, Sadek, F., and Simiu, E. (2002). Peak non-gaussian wind effects for database-assisted low-rise building design. J. Eng. Mech., ASCE, 128(5), The Mathworks, Inc., MATLAB the language of technical computing. Thomas, G., Sarkar, P. P., and Mehta, K. C. (1995). Identification of admittance functions for wind pressures from full-scale measurements. Proc. Ninth Int. Conf. Wind Eng., New Delhi, India, Tognarelli, M. A., Zhao, J., and Kareem, A. (1997). Equivalent statistical cubicization for system and forcing nonlinearities. J. Eng. Mech., ASCE, 123(8), TPU, Aerodynamic database of low-rise building, the 21 st Center Of Excellence program, Tokyo Polytechnic University, Winterstein, S. R. (1988). Nonlinear vibration models for extremes and fatigue. J. Eng. Mech., ASCE, 114(10), Winterstein, S. R., Ude, T. C., and Kleiven, G. (1994). Springing and slow-drift responses: predicted extremes and fatigue vs. simulation. Proc. BOSS-94, 3, MIT, Winterstein, S. R., and Kashef, T. (2000). Moment-based load and response models with wind engineering applications. J. Solar Energy Eng., ASME, 122,